A private network is pretty much a network owned specifically for use for a person or business. I guess you can use a law firm as an example. There are 5 offices and in each office is a computer connected to each other via the company network. The sole purpose of that network is used for looking up clients, cases, and to perform other tasks related to the job. Now more specifically, this network will most likely be a VPN or virtual private network. What makes this a virtual network? This is a VPN because attorneys working at home or on their break on a laptop can access the private network securely through a separate internet connection.
The benefits to using a VPN for company needs is that it encrypts all incoming and outgoing data so that it is not easily retrieved by someone who should not have access to the information. It has been discovered, usually long after the incident has occurred that someone has hacked into a network and was able to retrieve all outgoing and incoming data. This data can, and usually, contain personal and extremely confidential information. So it is wise to take the extra step in securing the information.
A con to using a private network is that access levels need to be granted to certain people. If a problem happens, which they sometimes do, you will most likely not be able to contact the secretary to reveal the information needed because they are warned ahead of time not to, especially, if it is a highly secured network. Another con to using a privatized network is the bandwidth allowed which is always regulated and may not be able to do much during non-business hours. Also, all traffic is usually regulated, bandwidth limits are set, and can get extremely costly during maintenance compared to using a public network.